Type Reduction Techniques for Two-dimensional Interval Type-2 Fuzzy Sets

In this paper, we address the issue of type reduction of multi-dimensional interval type-2 (IT2) fuzzy sets (FSs). We utilize the Karnik-Mendel (KM) algorithm to estimate the centroid boundary of a multi-dimensional footprint of uncertainty (FOU). We deal with two-dimensional (2-D) fuzzy sets as we can visualize the FOU using 3-D plots, thus making the illustration of the methods simple. However, the basic idea can be extended to multiple dimensions. We give a formal definition of the centroid boundary of a 2-D IT2 fuzzy membership function (FMF) and propose two methods for its estimation. The first method computes embedded type-1 (T1) FSs whose centroids constitute the centroid boundary. We obtain the embedded sets by producing slices of the domain using different sets of parallel planes and then apply the KM algorithm over each slice, to obtain "embedded-curves." For the second method, we approximate our first method by restricting embedded-curves to be "embedded-lines" thus enhancing computational speed. These type reduction techniques can be applied to applications involving multi-dimensional centroid estimation such as, clustering, support vector estimation for dimensionality reduction, fuzzy logic controllers, to mention a few.